Now we consider the Karhunen-Loeve Transform (KLT) (also known as Hotelling
Transform and Eigenvector Transform), which is closely related to the
Principal Component Analysis (PCA) and widely used in data analysis in many
fields.

Let be the eigenvector corresponding to the kth eigenvalue
of the covariance matrix , i.e.,

or in matrix form:

As the covariance matrix
is Hermitian
(symmetric if is real), its eigenvector 's are orthogonal:

and we can construct an unitary (orthogonal if is real)
matrix

satisfying

The eigenequations above can be combined to be expressed as:

or in matrix form:

Here is a diagonal matrix
. Left multiplying
on both sides,
the covariance matrix can be diagonalized:

Now, given a signal vector , we can define a unitary (orthogonal if
is real) Karhunen-Loeve Transform of as:

where the ith component of the transform vector is the projection of
onto :

Left multiplying
on both sides of the transform
, we get the inverse transform:

We see that by this transform, the signal vector is now expressed in an
N-dimensional space spanned by the N eigenvectors ()
as the basis vectors of the space.